The original deshima-sensitivity code follows the following flowchart

old deshima code

The optical chain described in the flowchart consists of serial transmission calculations from 1 medium to another. This entire chain of stacked transformations of the spectral input power can be regarded as a single affine transformation of the input and is independent of the bandwith or amount of frequency bins. It will remain unchanged throughout the proposed modifications.

The power impinging on a detector for a specific filter bin is calculated by taking the spectral flux density at the center frequency $\nu_c$ and multiplying this with the bandwith $\Delta\nu$. This matches the model of the previous described box filter, assuming the flux density $\mathrm{psd}$ is flat over the entire bandwith. This is not the case however, as we shall see.

Filter distribution

We know the $\mathrm{NEP}$ is proportional to the power received by the detector, and we have also seen that the atmospheric loading is a much bigger contribution to the spectral flux density than any atmospheric source. let's take a look at how this atmospheric loading varies over the bin width. In the figures below the actual frequency channels from DESHIMA 2.0 are taken and compared to ASTE atmospheric model from deshima-sensitivity.

For the channels where the atmospheric loading is reasonably constant over the bandwidth this approximation is fine, however when the spectral flux density peaks to twice it's local background level this is obviously a poor approximation for reality. Even if the filters were box filters, sampling at the center point and multiplying by the bandwidth can vary wildly from the actual box integration. This problem is even worse when we look at Lorentzian filters.

The peak at $286.1\:\mathrm{GHz}$ is obviously loading channel 131 and 132, but due to the wide profile of a Lorentzian even 130 and 133 are receiving energy from the peak in flux density.

A solution

The total power impinged on the detector can be approximated by calculating the spectral power arriving at the detector not just at the center frequencies $\overrightarrow{\nu_c}$ of the bins, but expanding the number of frequencies calculated to some amount of integration bins and calculating the spectral power at these frequencies. for a frequency $\nu_i$ and a channel $j$, the effective power loading by that frequency on that channel is given by:

$$ P_{\mathrm{KID},i,j} = \eta\left(\nu_i,j\right)\mathrm{psd}\left(\nu_i\right)\Delta_i\label{p_kid_i} $$

with $\Delta_i$ the bandwidth for that frequency. The total power impinged on a filter channel $P_j$ is then given by:

$$ P_{\mathrm{KID},j}=\sum_i P_{\mathrm{KID},i,j} = \sum_i \eta\left(\nu_i,j\right)\mathrm{psd}\left(\nu_i\right)\Delta_i \label{p_kid} $$

To make this more visual, take a look at the figure below

The calculated flux spectrum arriving at the filter will be calculated as is done in the current version of the deshima-sensitivity package, but using a finer frequency resolution. This spectrum will then arrive at the filter, where for each channel a weighted sum will be taken over all integration bins, resulting in a total power impinged on the detector.

Noise Equivalent Power

As seen, the photon noise equivalent power for a single DESHIMA channel $\mathrm{NEP}_{\tau=0.5\mathrm{s},\mathrm{ph},j}$is given by{cite %zmuidzinas_2003%}:

$$ \mathrm{NEP}_{\tau=0.5\mathrm{s},\mathrm{ph},j}^2=2\int_0^\infty\left(h\nu\right)^2\eta(\nu)n(\nu)\left[1+\eta(\nu)n(\nu)\right]d\nu $$

Again taking the filter efficiency $\eta\left(\nu_i,j\right)$ and photon number $n$ constant over a single integration bin and equating the photon number by

$$ n = \frac{P_\mathrm{KID}}{\eta h\nu\Delta\nu} $$

, we can easily take the Riemann sum of this integral as

$$ \mathrm{NEP}_{\tau=0.5\mathrm{s},\mathrm{ph},j}^2=2\sum_i\left( h\nu_iP_{\mathrm{KID},i}+\frac{P_{\mathrm{KID},i}^2}{\Delta\nu_i}\right)\label{nep_ph} $$

The generation-recombination noise $\mathrm{NEP_{GR}}$ is given by

$$ \mathrm{NEP}_{\mathrm{GR},j}^2=4\Delta_\mathrm{Al}\frac{P_{\mathrm{KID},j}}{\eta_\mathrm{pb}}=4\Delta_\mathrm{Al}\frac{\sum_i{P_{\mathrm{KID},j,i}}}{\eta_\mathrm{pb}} $$

Which results in a total $\mathrm{NEP}_{\tau=0.5\mathrm{s}}$ of

$$ \mathrm{NEP}_{\tau=0.5\mathrm{s}} = \sqrt{2\sum_i\left( h\nu_iP_{\mathrm{KID},i}+\frac{P_{\mathrm{KID},i}^2}{\Delta\nu_i}\right) + 4\Delta_\mathrm{Al}\frac{\sum_i{P_{\mathrm{KID},j,i}}}{\eta_\mathrm{pb}}}\label{total_NEP} $$

Photon Bunching

But what about photon bunching. I have shown that the coherence time and the bandwidth are inversely proportional. By decreasing the size of the integration bins we have also decreased the bandwidth and therefore increased the coherence time. Thankfully the integration time of our definition of the noise equivalent power, $0.5\:\mathrm{s}$, is orders of magnitude bigger than the coherence time [1]. As we have seen earlier, if the photons are sampled at orders of magnitude higher than the coherence time, doubling the coherence time (halving the bandwith) has no effect. You might say we are hiding the non-stochastic effects of photon bunching in the coarseness of our Riemann sum.

The Filter Matrix

From eq \eqref{p_kid_i} we can see that we need a two dimensional function $\eta\left(\nu_i,j\right)$ to go from a one dimensional $\mathrm{psd}$ to a power spectrum for each filter channel. This filter is made from Lorentzian profiles: $$ f(x)=\frac{A}{1+\left(\frac{x-x_0}{\gamma}\right)^2} $$

Where $\gamma$ is the half-width at half maximum (HWHM) value, for DESHIMA given as $$ \gamma=\frac{\nu_c}{2R} $$ with $R=500$.

Shown in the figure below is a visualization of how this filter matrix might look like.

With this framework in place it is easy to swap out the generated filter with another, more realistic model of the filters. The figure below shows a simulated filter transmission curve for the DESHIMA 2.0 chip.

DESHIMA 2.0 filter-profile A simulated filter profile of the DESHIMA 2.0 spectrometer

As you can see, the filters aren't all equally efficient and wide. In the figure below all 346 bins and the corresponding values of $R$ are plotted, with the size corresponding to the transmission.

While using a perfect Lorentzian approximation for the filter profiles for each channel is better than the center frequency sampling that is done now, it might be more advantageous to be able to use simulated or measured profiles. The filter matrix will therefore, depending on the users choice, be either generated via Lorentzian curves or loaded in via a file.

Transforming the calculated noise

Once the $\mathrm{NEP}$ has been calculated, it needs to be transformed back into something more usable to define the sensitivity of the instrument. We do this by calculating the source coupling and a quantity named the Noise Equivalent Source Flux $\mathrm{NEF}$. The $\mathrm{NEF}$ is defined as the amount of flux a source needs to emit to equal the $\mathrm{NEP}$ in strength and is therefore our sensitivity: a source with a flux lower than the $\mathrm{NEF}$ will be hidden in the noise of the measurement data. The $\mathrm{NEF}$ is given by: $$ \mathrm{NEF} = \frac{\mathrm{NESP}_{\tau=0.5\mathrm{s}}}{\sqrt{2}A_g}=\frac{\mathrm{NEP_{inst,{\tau=0.5\mathrm{s}}}}}{\eta_\mathrm{sw}\sqrt{2}A_g}=\frac{\mathrm{NEP}_{\tau=0.5\mathrm{s}}}{\eta_\mathrm{inst}\eta_\mathrm{sw}\sqrt{2}A_g} $$

with $A_g$ the area of the telescope, $\eta_\mathrm{sw}$ the aggregate efficiency from the source to the window of the cryostat chamber DESHIMA is housed in and $\eta_\mathrm{inst}$ the instrument efficiency. The first two remain unchanged throughout the modifications discussed in this chapter, however $\eta_\mathrm{inst}$ is dependent on $\eta_\mathrm{filter}$ and is therefore modified.

Since the calculation of the $\mathrm{NEP}_{\tau=0.5\mathrm{s}}$ as in \eqref{total_NEP} collapses the noise equivalent power down to a single value per channel, we can approximate our source flux as a scalar per channel too. In order to have this approximation hold we need to consider two separate cases: a spectral source and a continuum source.

A Spectral Source

A spectral source is defined as an abject that transmits only within the bandwidth of a single filter channel, where we define the bandwidth as the $\mathrm{FWHM}$

In this case the box-filter approximation holds perfectly fine

The total transmission of the box filter should be such that it matches the transmission of the Lorentzian profile in the bandwidth:

$$ \int_{\nu_0-\gamma}^{\nu_0+\gamma}\frac{A\gamma^2}{\left(\nu-\nu_0\right)^2+\gamma^2}d\nu=2\gamma\cdot h\Leftrightarrow h=\frac{\pi}{4}A $$

meaning that for spectral sources we can set the $\eta_\mathrm{filter}$ component of $\eta_\mathrm{inst}$ to a $\pi/4$ times the peak value of the Lorentzian filter.

A Continuum source

Besides spectral sources, astronomical objects also radiate a continuous spectrum, through blackbody radiation:

Because this continuum source is as near as makes no difference constant over the frequency bin, we can model it's transmission by another box-filter. If we set the height to the same $\pi/4A$, we can calculate the extra area and thus power received of the out of band incoming power as follows: $$ \int_{-\infty}^{\infty}\frac{A\gamma^2}{\left(\nu-\nu_0\right)^2+\gamma^2}d\nu=\pi\gamma A=B\cdot 2\gamma\cdot\frac{\pi}{4}A\Leftrightarrow B=2 $$ meaning that the instrument is exactly twice as sensitive to continuum flux than it is to spectral flux.

The Final Model

Since both sources are approximated well using a box filter, for the purposes of calculating the $\eta_\mathrm{inst}$, the filter efficiency is simply modeled as a box filter with height $A\pi/4$ and with width $\mathrm{FHWM}$ or $2\mathrm{FHWM}$ for a spectral and a continuum source respectively. This means that the total model looks like this: new deshima code